Notes on stochastic integration theory with respect to c\`adl\`ag semimartingales and a brief introduction to L\'evy processes

TL;DR

These notes provide an accessible overview of semimartingale theory and stochastic integration with respect to cdlg semimartingales, emphasizing their application to Le9vy processes and stochastic differential equations.

Contribution

They offer a foundational, proof-free introduction to semimartingales and stochastic integration, highlighting differences from continuous cases and focusing on Le9vy-driven equations.

Findings

Clarifies the fundamentals of cdlg semimartingales

Discusses advantages and limitations of stochastic integration in this setting

Introduces Le9vy processes and their role in stochastic differential equations

Abstract

The purpose of these notes is to distribute, mostly without proofs, fundamental definitions and results concerning the theory of semimartingales and stochastic integration. The material serves as a foundational guide for those interested in applying these concepts, particularly in the study of stochastic (functional) differential equations driven by L\'evy processes. These notes are adapted from the preliminary chapter of the author's master's thesis (with only minor changes) and are intended to introduce newcomers to the essentials of c\`adl\`ag semimartingale theory while also discussing the advantages, limitations, and subtleties as compared to stochastic integration in the continuous setting.